On the other hand, in the presence of the lattice relaxation, we need to include more shells to achieve the comparable accuracy. This is demonstrated in the Fig. REF . The increase of needed shells results from the factor \(e^{i (G + K)\cdot (U_{j, S}^{\parallel }(x) - U_{j^{\prime }, S^{\prime }}^{\parallel }(x))}\)  in Eq. REF . The Fourier transform of the inter-layer hopping (i.e. for \(j \ne j^{\prime }\) ) is the largest for \(G\) s satisfying \(|G + K| = |K|\) . Because of the spatial inhomogeneity of the lattice relaxation \(\delta U\)  (Eq. REF ), the mentioned exponential factor induces the inter-layer scattering with the momentum transfer of all possible \(q_{\mu , l}\) ; the strength of the scattering is proportional to the Fourier transform of the exponential factor. For the lattice relaxation in Ref. [1]}, \(\delta U\)  is dominated by the lowest wavevectors \(\pm g_1\) , \(\pm g_2\)  and \(\pm (g_1 + g_2)\) . As a consequence, \(H_{inter}\)  should include, at least, the terms with the momentum transfer of \(q_{1, l} \pm g_1\) , \(q_{1, l} \pm g_2\) , and \(q_{1, l} \pm (g_1 + g_2)\) , i.e. all the \(q\)  vectors in the first three shells. While all the values of \(w_i^{(\mu , l)}\)  and \(\lambda _i^{(\mu , l)}\)  in the first three shells can be obtained from Table REF  and the formula listed in Eq. REF –REF , the values in the next three shells can be calculated in the same way based on Table REF  and Eq. REF –REF  in the appendix. Fig. REF (d) shows that the first four shells are needed to achieve the accuracy of \(0.3\) meV at \(\Gamma \)  for the lattice configuration of Ref. [1]}.
